1. W E B E X T E N S I O N 28A
Derivation of Annuity Formulas
Following are derivations for annuity formulas. It is actually easier to start with the
formula for a perpetuity. First, consider the following geometric progression, where
A is a positive constant that is less than 1, and X is the sum of the geometric
progression:
q
X = a At (28A-1)
t=1
At first blush, it might seem that X must equal infinity, since the sum goes to
infinity. But notice that as t gets large, the term At gets very small, because A is less
than 1. For example, suppose that A is equal to 1⁄2. Then the sum is
X = 1>2 + 1>4 + 1>8 + 1>16 + Á
Notice that as we continue to add terms, X approaches but never exceeds 1.
Perhaps from an algebra or calculus course you recall that the sum of a geometric
progression actually has a closed-form solution:
A
q
X = a At = (28A-2)
t=1 1 - A
For example, in the case of A ϭ 1⁄2, the sum is
q
1>2
X = a (1>2)t = = 1 (28A-3)
t=1 1 - (1>2)
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2. 28A-2 • Web Extension 28A Derivation of Annuity Formulas
Now consider a perpetuity with a constant payment of PMT and an interest rate
of I. The present value of this perpetuity is
PMT
q
PV = a (28A-4)
t=1 (1 + I)t
This can be written as a geometric progression:
t
= PMTa a b
1 1
q q
PV = PMTa t
(28A-5)
t = 1 (1 + I) t=1 1 + I
Because 1 ϩ I is positive and greater than 1 for reasonable values of I, the summa-
tion in Equation 28A-5 is a geometric progression with A ϭ 1͞(1 ϩ I). Therefore,
using Equation 28A-2, we can write the summation in Equation 28A-5 as
b a
1
t
1 + I
a b =
1 1
q
a 1 + I = (28A-6)
a1 - a bb
t=1 1 I
1 + I
Substituting this result into Equation 28A-4 gives us the present value of a perpetuity:
PMT
PV = (28A-7)
I
Now consider the time lines for a perpetuity that starts at time 1 and a perpe-
tuity that starts at time N ϩ 1:
0 1 2 N Nϩ1 Nϩ2 Nϩ3
___________________ . . . _________________________________ . . .
PMT PMT PMT PMT PMT PMT
0 1 2 N Nϩ1 Nϩ2 Nϩ3
____________________ . . . ________________________________ . . .
PMT PMT PMT
Notice that if we subtract the second time line from the first, we get the time line for
an ordinary annuity with N payments:
0 1 2 N
____________________ . . . ___
PMT PMT PMT
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3. Derivation of Annuity Formulas • 28A-3
Therefore, the present value of an ordinary annuity is equal to the present value of
the first time line minus the present value of the second time line. The present value
of the first time line, which is a perpetuity, is given by Equation 28A-7
PMT
PV of first time line = (28A-8)
I
If we apply Equation 28A-7 to the second time line, it gives the value of the pay-
ments discounted back to time N (because if we just look at the time line from N
on, it is an ordinary annuity that starts at time N ϩ 1). To find the present value of
the second time line, we just discount this perpetuity value back to time 0:
PV of second time line = a b
PMT 1
(28A-9)
I (1 + I)N
Subtracting Equation 28A-9 from 28A-8 gives the present value of an ordinary
annuity, PVA:
PVA = a b - a b
PMT PMT 1
(28A-10)
I I (1 + I)N
This can be rewritten as
PVA = PMT c a b - a bd
1 1
(28A-11)
I I(1 + I)N
The future value of an ordinary annuity is equal to the present value
compounded out to N periods:
FVA = PVA(1 + I)N = PMT c a b - a b d(1 + I)N
1 1
(28A-12)
I I(1 + I)N
This can be rewritten as
(1 + I)N
FVA = PMT c a b - a bd
1
(28A-13)
I I
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